Abstract
Coplanar Al/graphene/Al junctions fabricated on the same graphene sheet deposited on silicon carbide (SiC), show robust Josephson coupling at subKelvin temperature, when the separations between the electrodes is below 400 nm. Remarkably, a hysteretic Critical State sets in when ramping an orthogonal magnetic field, with a sudden collapse of the Josephson critical current Ic when turning the field on, and a revival of Ic when inverting the sweep. Similar hysteresis can be found in granular superconducting films which may undergo the Berezinskii-Kosterlitz-Thouless transition. Here, we give quantitative arguments to prove that this odd behavior of the magnetoconductance gives evidence for an incipient Berezinskii-Kosterlitz-Thouless transition with drift and pinning of fluctuating free vortices induced by the current bias.